| L(s) = 1 | + 3-s + 2·5-s + 4·11-s − 4·13-s + 2·15-s + 6·17-s + 4·19-s + 5·25-s − 27-s − 4·29-s + 4·33-s − 6·37-s − 4·39-s + 4·41-s + 8·43-s + 6·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s + 2·61-s − 8·65-s + 4·67-s + 6·73-s + 5·75-s − 16·79-s − 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.20·11-s − 1.10·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 25-s − 0.192·27-s − 0.742·29-s + 0.696·33-s − 0.986·37-s − 0.640·39-s + 0.624·41-s + 1.21·43-s + 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.992·65-s + 0.488·67-s + 0.702·73-s + 0.577·75-s − 1.80·79-s − 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.695276782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.695276782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.141586222502462082609396679405, −8.986110837635124187554553854430, −8.525585438477936160443132809337, −7.912485992073285104694939895389, −7.60716961882873176513262596023, −7.39183450447668903519168005615, −6.85300057974494424073083374816, −6.54951996829193710377981229934, −5.90441797912309754295200006364, −5.79227885712163690276153584226, −5.07172018458693842342994873903, −5.04949071521281981685592426534, −4.36813759831279827187110938651, −3.69485898387662226937709894375, −3.45972651216623040377621903417, −3.02200297793579465275070352352, −2.24281320202841076040231901051, −2.10471627684087977317770597999, −1.25472338647344124511488483847, −0.78354883221851688463399060692,
0.78354883221851688463399060692, 1.25472338647344124511488483847, 2.10471627684087977317770597999, 2.24281320202841076040231901051, 3.02200297793579465275070352352, 3.45972651216623040377621903417, 3.69485898387662226937709894375, 4.36813759831279827187110938651, 5.04949071521281981685592426534, 5.07172018458693842342994873903, 5.79227885712163690276153584226, 5.90441797912309754295200006364, 6.54951996829193710377981229934, 6.85300057974494424073083374816, 7.39183450447668903519168005615, 7.60716961882873176513262596023, 7.912485992073285104694939895389, 8.525585438477936160443132809337, 8.986110837635124187554553854430, 9.141586222502462082609396679405
